Duke Mathematical Journal

On a refinement of Waring's problem

Van H. Vu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J., Volume 105, Number 1 (2000), 107-134.

First available in Project Euclid: 13 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11P05: Waring's problem and variants
Secondary: 05D40: Probabilistic methods 11P55: Applications of the Hardy-Littlewood method [See also 11D85]


Vu, Van H. On a refinement of Waring's problem. Duke Math. J. 105 (2000), no. 1, 107--134. doi:10.1215/S0012-7094-00-10516-9. https://projecteuclid.org/euclid.dmj/1092403818

Export citation


  • S. L. G. Choi, P. Erdős, and M. Nathanson, Lagrange's theorem with $N^1/3$ squares, Proc. Amer. Math. Soc. 79 (1980), 203--205.
  • P. Erdős, ``Problems and results in additive number theory'' in Colloque sur la Théorie des Nombres (Bruxelles, 1955), Masson and Cie, Paris, 1956, 127--137.
  • P. Erdős and M. Nathanson, ``Lagrange's theorem and thin subsequences of squares'' in Contribution to Probability, ed. J. Gani and V. K. Rohatgi, Academic Press, New York, 1981, 3--9.
  • P. Erdős and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85--90.
  • P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arith. 6 (1960), 83--110.
  • P. Erdős, A. Sárközy, and V. T. Sós, Problems and results on additive properties of general sequences, III, Studia Sci. Math. Hungar. 22 (1987), 53--63.
  • P. Erdős and P. Tetali, Representations of integers as the sum of $k$ terms, Random Structures Algorithms 1 (1990), 245--261.
  • P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212--215.
  • H. Halberstam and K. F. Roth, Sequences, 2d ed. Springer, New York, 1983.
  • D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl unter Potenzen (Waringsche Problem), Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse aus dom Jahr 1909, 17--36.; Math. Ann. 67 (1909), 281--300.
  • S. Janson, Poisson approximation for large deviations, Random Structures Algorithms 1 (1990), 221--229.
  • J. H. Kim and V. H. Vu, Concentration of multivariate polynomials and its applications, Combinatorica 20 (2000), 417--434.
  • --------, Small complete arcs on projective planes, submitted.
  • M. Kolountzakis, ``Some applications of probability to additive number theory and harmonic analysis'' in Number Theory (New York, 1991--1995.), ed. D. V. Chudnovsky, G. V. Chudnovsky, and M. B. Nathanson, Springer, New York, 1996, 229--251.
  • M. Nathanson, ``Waring's problem for sets of density zero'' in Analytic Number Theory (Philadelphia, 1980), ed. M. Knopp, Lecture Notes in Math. 899, Springer, Berlin, 1981, 301--310.
  • --------, Additive Number Theory: The Classical Bases, Grad. Texts in Math. 164, Springer, New York, 1996.
  • C. Pomerance and A. Sárközy, ``Combinatorial number theory'' in Handbook of Combinatorics, Vol. 1, ed. R. Graham, M. Grötschel, and L. Lovász, Elsevier,Amsterdam, 1995, 967--1018.
  • I. Ruzsa, A just basis, Monatsh. Math. 109 (1990), 145--151.
  • B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959), 1--96.
  • S. Sidon, Ein Satz über trigonomische Polynome und seine Anwendung in der Theorie der Fourier-Reihen, Math. Ann. 106 (1932), 536--539.
  • J. Spencer, ``Four squares with few squares'' in Number Theory (New York, 1991--1995.), ed. D. V. Chudnovsky, G. V. Chudnovsky, and M. B. Nathanson, Springer, New York, 1996, 295--297.
  • M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), 1--34.
  • R. C. Vaughan, The Hardy-Littlewood Method, Cambridge Tracts in Math. 80, Cambridge Univ. Press, Cambridge, 1981.
  • V. H. Vu, On some simple degree conditions that guarantee the upper bound on the chromatic (choice) number of random graphs, J. Graph Theory 31 (1999), 201--226.
  • --. --. --. --., On the concentration of multivariate polynomials with small expectation, Random Structures and Algorithms 16 (2000), 344--363.
  • --. --. --. --., New bounds on nearly perfect matchings of hypergraphs: Higher codegrees do help, Random Structures and Algorithms 17 (2000), 29--63.
  • --------, Some new concentration results and applications, manuscript.
  • E. Wirsing, Thin subbases, Analysis 6 (1986), 285--308.
  • J. Zöllner, Über eine Vermutung von Choi, Erdős und Nathanson, Acta Arith. 45 (1985), 211--213.
  • --------, Der Vier-Quadrate-Satz und ein Problem von Erdős and Nathanson, Ph.D. thesis, Johannes Gutenberg-Universität, Mainz, 1984.